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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"8c9e06eeac8177c16390ef4ca2caac942ca4e1f92b0e951d20dc31e5abf6983a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8f767a3615460a6b","role":"sequence","name":"Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      2,       2;\n  2:      5,      10,       3;\n  3:     15,      46,      30,       4;\n  4:     52,     218,     228,      76,       5;\n  5:    203,    1088,    1593,     896,     180,       6;\n  6:    877,    5752,   10923,    8816,    3155,     414,      7;\n  7:   4140,   32226,   75624,   80356,   42355,   10494,    938,      8;\n  8:  21147,  190990,  535692,  710524,  500855,  187986,  33866,   2104,    9;\n  9: 115975, 1194310, 3907353, 6239152, 5547365, 2830962, 796649, 107632, 4680, 10;\n","recurrence_sentence":"The row polynomials satisfy -(x-n*x^3)*P(n,x) = (2*n+(n^2+5*n-3)*x-4*x^2-(n^3+5*n^2+3*n-3)*x^3-(2*n^3+4*n^2-2*n-4)*x^4-(n^3-n)*x^5)*P(n-2,x) - (1+(n+4)*x+(2*n+4)*x^2-(n^2+2*n+2)*x^3-(3*n^2+7*n+5)*x^4-(3*n^2+5*n+2)*x^5-(n^2+n)*x^6)*d/dx P(n-2,x) - (2+(2*n+3)*x-2*x^2-(2*n^2+4*n+2)*x^3-2*n^2*x^4)*P(n-1,x) + (x+2*x^2-(n-1)*x^3-2*n*x^4-n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*P[n - 2, x] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*P[n - 1, x] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - n*x^5)*D[P[n - 1, x], x])/(-x + n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + (n^2 + 5*n - 3)*x - 4*x^2 - (n^3 + 5*n^2 + 3*n - 3)*x^3 - (2*n^3 + 4*n^2 - 2*n - 4)*x^4 - (n^3 - n)*x^5)*v[i - 2] - (1 + (n + 4)*x + (2*n + 4)*x^2 - (n^2 + 2*n + 2)*x^3 - (3*n^2 + 7*n + 5)*x^4 - (3*n^2 + 5*n + 2)*x^5 - (n^2 + n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 3)*x - 2*x^2 - (2*n^2 + 4*n + 2)*x^3 - 2*n^2*x^4)*v[i - 1] + (x + 2*x^2 - (n - 1)*x^3 - 2*n*x^4 - n*x^5)*deriv(v[i - 1], x))/(-x + n*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","\u03b8+1"],"edges":["e-236d0fb15dc6eca0","e-9b9aa33d8b05d5b3"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 \u03b8+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 \u03b8+1","chain":null}